Computing arc length in Poincare disk model of hyperbolic space

I am reading Thurston's book on the Geometry and Topology of 3-manifolds, and he describes the metric in the Poincare disk model of hyperbolic space as follows:

... the following formula for the hyperbolic metric ds^2 as a function of the Euclidean metric x^2:

[tex]ds^2 = \frac{4}{(1-r^2)^2} dx^2[/tex]

I don't understand how what ds^2 means or how to use this formula to compute distances and arc lengths. A naive guess is that the arc length should be given by

[tex]s = \int_a^b \sqrt{ds^2}[/tex]

but that doesn't seem to give me the correct answer. For example, take a point with Euclidean distance r from the origin. What is its distance in the hyperbolic metric?

I know that the distance should be the arc length of the straight line connecting 0 to x, since the straight line through the origin is a geodesic in the hyperbolic metric. My guess would give me an arc length of

[tex]s = \int_0^r \frac{2}{1-x^2} dx = 2 arctanh(r)[/tex]

However, another website claims that the answer should be log(1+r)/log(1-r).

There are some mistakes in your claims.
First of all the expression for the metric cannot be "square-rooted" so easily, in fact
[tex] \mathrm{d}x^2 = \mathrm{d}x^i\mathrm{d}x_i = \mathrm{d}x^2+\mathrm{d}y^2 [/tex]
then the definition of arc lenght is the integral of the velocity along the curve (i.e. the trajectory), defined as (be careful there can be a sign under the square root, it depends on your convention):
[tex] L = \int^b_a \sqrt{\frac{\mathrm{d}x^{\mu}}{\mathrm{d}t} \frac{\mathrm{d}x^{\nu}}{\mathrm{d}t} g_{\mu\nu}} \mathrm{d}t [/tex]
with [tex] x^{\mu} = \left\{x,y\right\} [/tex] .
So the first thing you have to do is to choose a parametrization of your coordinates and then proceed to compute the integral. Note also that
[tex] r^2 = x^2 + y^2 [/tex]
Let me know if you get the right result! ;)

Thanks, but I'm not really sure what to do with the information you just gave me. What does [tex]g_{\mu\nu}[/tex] mean? Do I really need to compute the Christoffel symbols? If yes, how would I compute them from "ds^2"?

All I care about right now is the simplest way to get from "ds^2" to arc length. If I don't absolutely need to compute all the other crap, I'd rather not.